For the case of plane polar co-ordinates \(r, \theta\), write the unit vectors \(e_{r}\) \((=\hat{\boldsymbol{r}})\) and \(\boldsymbol{e}_{\theta}\) in terms of \(\boldsymbol{i}\) and \(\boldsymbol{j}\). Hence show that \(\partial \boldsymbol{e}_{r} / \partial \theta=\boldsymbol{e}_{\theta}\) and \(\partial \boldsymbol{e}_{\theta} / \partial \theta=-\boldsymbol{e}_{r} .\) By starting with \(\boldsymbol{r}=r e_{r}\) and differentiating, rederive the expressions for the components of the velocity and acceleration vectors.

Plane polar coordinates offer a different framework from the traditional Cartesian coordinate system for locating points on a two-dimensional plane. In this system, any point can be described by just two values: the radial distance from a central point (commonly referred to as the origin) and the angular displacement from a reference direction.

In terms of mathematics, if we have a point P, its location in plane polar coordinates is given by a pair \( (r, \theta) \), where \( r \) is the distance of P from the origin (which can be thought of as the length of the straight line connecting the origin to P), and \( \theta \) is the angle measured from the positive x-axis to the line segment OP in a counter-clockwise direction.

Polar coordinates are extremely helpful in scenarios where problems have a natural structure of radial symmetry, such as the motion of planets around the sun or the ripples from a stone thrown into a pond. In classical mechanics, they simplify calculations involving paths that are circular or radial.

In the polar coordinate system, unit vectors play an integral role in describing directions with respect to points in the plane. Unlike the constant unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) of the Cartesian coordinate system, unit vectors in the polar coordinates vary with position.

The radial unit vector, \(\mathbf{e}_r\), always points directly away from the origin and is parallel to the radius connecting the origin to the point. For a given angle \(\theta\), it is expressed as \(\mathbf{e}_r = \cos{\theta}\mathbf{i} + \sin{\theta}\mathbf{j}\).

The angular unit vector, \(\mathbf{e}_\theta\), is perpendicular to the radial unit vector, pointing in the direction of increasing \(\theta\). It is described by the equation \(\mathbf{e}_\theta = -\sin{\theta}\mathbf{i} + \cos{\theta}\mathbf{j}\).

The beauty of these vectors is their locality – they change orientation as \(\theta\) changes, ensuring that \(\mathbf{e}_r\) always points radially outwards and \(\mathbf{e}_\theta\) tangentially to the circle at that radius.

For analysis in classical mechanics, we often need to explore how unit vectors change as we move around in the polar coordinate system. This change is quantified by taking the partial derivatives of \(\mathbf{e}_r\) and \(\mathbf{e}_\theta\) with respect to \(\theta\).

The partial derivative of the radial unit vector \(\mathbf{e}_r\) with respect to \(\theta\) gives the angular unit vector \(\mathbf{e}_\theta\) as shown in the equation \(\frac{\partial \mathbf{e}_r}{\partial \theta} = -\sin{\theta}\mathbf{i} + \cos{\theta}\mathbf{j} = \mathbf{e}_\theta\). Similarly, when we take the partial derivative of \(\mathbf{e}_\theta\), it results in the negative radial unit vector \(\frac{\partial \mathbf{e}_\theta}{\partial \theta} = -\cos{\theta}\mathbf{i} - \sin{\theta}\mathbf{j} = -\mathbf{e}_r\).

The implications of these results are profound in dynamics, where such derivatives help us describe rotational motion and the resulting effects on objects moving in a curved path.

In classical mechanics, understanding an object's state of motion involves analyzing its velocity and acceleration. In polar coordinates, these vectors have a unique representation due to the coordinate system's nature.

The velocity vector in polar coordinates can be found by differentiating the position vector \(\mathbf{r}\) with respect to time. This results in two components: the radial velocity (change of radius with time) \(\frac{\text{d} r}{\text{d} t}\mathbf{e}_r\) and the angular velocity (radius multiplied by the change of angle with time) \(r\frac{\text{d} \theta}{\text{d} t}\mathbf{e}_\theta\).

The acceleration vector is the time derivative of the velocity vector, encompassing changes in both magnitude and direction of the velocity. For acceleration, this yields a radial component, which includes both the centripetal acceleration (due to change in direction at constant radius) and radial acceleration (due to change in radius), and an angular component, which includes the Coriolis and angular accelerations. The expressions derived above provide us with an invaluable tool for analyzing motion in circular paths or any path that can be defined radially and angularly.

Understanding these components helps in solving many mechanics problems like the orbits of celestial bodies, projectiles launched at an angle, and objects experiencing rotational forces.